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Approximate Factorization of Multivariate Polynomialsand Absolute Irreducibility Testing

T. Sasaki, M. Suzuki, M. Kol\’a\v{r} and M. Sasaki

RIKEN (Institute ofPhysical and Chemical Research)Wako-shi, Saitama, 351-01, Japan

Abstract

A concept of approximate factorization of multivariate polynomials is introduced and an algorithm for ap-proximate factorization is presenoeA The algorithm handles polynomials with complex coefficients representedapproximately, hence it can be used to test the absolute irreducibility of $multiv\pi iate$ polynomials. The algorithmworks as follows: given a monic square-free polynomial $F(x,y, \ldots, z)$, it calculates the roots of $F(x, y0, \ldots, z_{0})$

numericaUy, where $y0,$ $\ldots,$ $z0$ are suitably chosen numbers, then it $\infty nsmlcts$ power series $F_{1},$$\ldots$ , $F_{n}$ such that

$F(x,y, \ldots , z)\equiv F_{1}(x, y, \ldots, z)\cdots F_{n}(x, y, \ldots, z)(mod S^{\epsilon+l})$ , where $n=\deg_{x}(F)$. $S=(y-y0, \ldots , z-zo)$ , and$e= \max\{\deg_{y}(F), \ldots , \deg_{z}(F)\}$ ; finaUy it flnds the approximate divisors of $F$ as products ofelements of $\{F_{1}, \ldots, F_{n}\}$ .

1 Introduction

Consider, for example, the polynomial

$F(x, y)=x^{3}+3.\omega x^{2}+(2.2y^{2}-1.7)x-2.9y$ .

Although $F(x, y)$ is irreducible, we can decompose it as$F(x, y)=(x^{2}+1.3yx-1.7)(x+1.7y)-O.01y^{2}x-O.Oly$ .

We call thhis kind of decomposition approximate factorization of $accuracy\sim 10^{-2}$.Factorization of polynomials has been studied by many persons. Some of the ol&r algorithms are described in

$[vdW37]$ . The more modern ones which are $P^{I\infty nca1}$ but have exponential time-complexity in the worst case weredevised by [Ber67, Zas69, WR75, WR76], etc. (see $[Kal82a]$ for rather complete biblioyaphy), and [LLL82, $Kal82b$,$vzGK83$ , Len84], etc., have presented algorithms of polynomial time-complexity. Furthermore, several authors havestudied factorization over an algebraically closed field and absolute irreducibility testing [HS81, CG82, Ka185, YNT90].However, all of these studies deal with the exact $fact\propto izanon$, hence the algorithms proposed are not applicable topolynomials with coefficients of floating-point numbers, for example.

On the other hand, scientists and engineers do often desire to”factorize” $mMnvariate$ polynomials with coefficientsof approximate numbers. According to their view, both $(x^{2}-10\alpha r)$ and $(x^{2}-\iota o\alpha D.5)$, for example, are almost thesame and “factorized” as

$(x^{2}-10t\mathfrak{M}.5)\simeq(x^{2}-1\mathbb{R})=(x+1\infty)(x-1\alpha))$.Thus, we are naturally led to extend the concept of factorization to the approximate one. This kind of extension isdesirable not only for factorization but also for many other algebraic operations. In fact, recently one of the authors(T.S.) and his collaborators developed algorithms for approximate GCD, univariate and multivariate, and applied themto solving numerically ill-conditioned problems [SN89, SS90, ONS]. They called such algorithm$s$ the approximatealgebraic algorithms which include algorithms for approximate factorization, $t\infty$ .

Since most of the conventional factorization algorithms employ the exact numerical arithmetic (to our knowledge,only [Len84] and [Ka185] employ the approximate numerical arithmetic), we cannot apply them to approximatefactorization, and we need a new idea. Our idea is to use the fact that monic square-free polynomial $F(x, y)$ has theroots $x=\varphi_{i}(y),$ $i=1,$ $\ldots,$ $\deg_{x}(F)$ , of the form $\varphi_{i}(y)=c_{i,0}+c_{i,1}y+c_{i,2}y^{2}+\cdots$ . We thought that this fact had not beenused in the conventional factorization algorithms, but we found that [Ka185] utilized this fact in a rather restricted way.However, we realized that the above-mentioned fact can be utilized more widely and eleganUy than as used in [Ka185].

After giving some preliminaries in 2, we reformulate in 3 the generalized Hensel construction which we will use tocalculate the roots of $F(x, y)$ in the form of fornal power series in $y$ . The principle of (approximate) factorization isproved in 4, and a primitive version of the algorithm is presented. The primitive version is, although complete, quiteinefficient unless the degree of the polynomial is low. So, in 5, we present a simple method for making the algorithmpractical.

数理解析研究所講究録第 746巻 1991年 59-68

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2 Preliminaries

In this section, we define notations, explain preprocessing of wlynomM. and inmuce notions on $a_{PN}ox\dot{m}a\ddagger e$

$faclo\dot{n}za\mathfrak{g}on$ by referring to [SN89, SS90].

$\Pi e$ field of complex numbers is denoted by C. By $C[x, y, \ldots, z]$ and $C\{y)\cdot. , z\}[x]$ we mean, res#ctively, thewlynomial ring over $C$ in variables $x,$ $y,$ $\ldots,$

$z$ . and the ring of polynomials in $x$ with coefficients of fornal power $s$eriesin $y,$ $\ldots,$

$z$ over C. In thhis paper, $x$ is $\alpha ealed$ as the main variable.Let $F$($x,$ $y,$ $\ldots$ , z) be an element of $C[x, y, \ldots, z]$ and

$F(x, y, \ldots, z)=f_{n}(- y, \ldots, z)x"+f_{n-1}(y, \ldots, z)x^{n-1}+\cdots+f_{0}(y, \ldots, z)$ , $f_{n}\neq 0$ (2.1)

The &gree of $F$ w.r.t. $x$ is denoted by $\ g_{x}(F)$ or simply by $\deg(F):\deg(F)=n$ . Similarly, the degree of $F$ w.r.t. $y$ isdenoted by $\deg_{y}(F)$. The leading $co\phi fi\dot{\alpha}ent$ of $F$ w.r.t. $x$ is $f_{n}$ and denoted by $1c.(F)$ or simply by $1c(F):1c(F)=f_{n}$ .Let $f$ be a $\mu lynom\mathfrak{M}wwer$ series in a single variable. By $\lfloor f\rfloor_{\epsilon}$ , we mean the sum of $aU$ the term$s$ , of $f$ , of degrees notless than $e$ ; if $f=a_{0}+a_{1}y+a_{2}y^{2}+\cdots$ , then $\lfloor f\rfloor_{e}=a_{\epsilon}y^{e}+a_{\epsilon+1}y^{e+1}+\cdots$.

Polynomial $F$ is caUed monzc w.r.$tx$ if $1c_{x}(F)=1$ . Any polynomial can be transformed into a monic polynomialby the foUowing ffansformation

$F(x, y, \ldots, z)arrow F’(x, y)\cdots$ , $z$ ) $=f_{n}^{n-1}F(x/f_{n)}y, \ldots , z)$. (2.2)

The $F$($x,$ $y,$ $\ldots$ , z) can be decomposed as$F(x, y, \ldots, z)=F_{m}^{m}(x, y, \ldots, z)F_{m-\iota)}^{m-\iota_{(X,y,\ldots z)\cdots F_{1}(x,y,\ldots,z)}}$,each $F_{1}$ has no multiple factor, $i=1,$ $\ldots m$) ’ (2.3)

$gcd(F_{\dot{*}}, F_{j})=1$ for $i\neq j$ .

This $d\infty om\varphi siaon$ is caUed square-free $d\ell composition$ and each $F_{i}$ is caUed square-free. Let $b_{y)}\ldots,$ $b_{z}$ be numbers inC. If we $ch\infty seb_{y},$

$\ldots,$$b_{z}$ arbiffarily then $F(x, b_{y}, \ldots, b_{z})$ is mostly square-free.

Remark 2.1 With the above preprocessing, we may assume without loss of generahty that $F$ ($x,$ $y,$ $\ldots$ , z) is monic andboth $F(x, y, \ldots, z)$ and $F(x, 0, \ldots, 0)$ are square-free. $0$

Definition 2.1 [manmm magnitude $co\phi ficient$]. The maximum magnitude numerical coefficient of $F$ is denoted by$mmc(F)$. ($mmc(F)$ is nothing but the infinity norm $|F|_{\infty}.$) $0$

Definition 2.2 [numbers of similar magnitude]. Let $a$ and $b$ be numbers in $C$ , and $b\neq 0$ . By $a=O(b)$, we mean that$1/c\leq|a/b|\leq c$ where $c\geq 1$ is apositive number not much different from 1. (Usually, $O$’ denotes Landau’s notation,and we are using it in a somewhat different meaning.) $0$

Definition 2.3 [nomalization ofpolynomial]. Normalization ofpolynomial $F$ is the scale aansformation $Farrow F’=\eta F$ ,$\eta\in C$ , so that $mmc(F’)=1.0$

Now, we define approximate $factoriza\dot{u}on$ .

Definition 2.4 [approximate factorization of accuracy $\epsilon$ ]. Let $\epsilon$ be a small positive number, $0<\epsilon\ll 1$ . Let$F$ ($x,$ $y,$ $\ldots$ , z) be a normalized polynomial in $C[x, y, \ldots, z]$ . If

$F=GH+\Delta F$, $mmc(\Delta F)=O(\epsilon)$ , (2.4)

where $G$ and $H$ are non-constant elements of $C[x, y, \ldots, z]$ , then $G$ (and $H$) is called an approximate divisor of $F$ ofaccuracy $\epsilon$ . Finding $G$ and $H$ satisfying (2.4) is the approximate factorization, of accuracy $e$ , of F. $0$

Remark2.2 If $F$ is not approximately factorizable with accuracy less than $O(\epsilon)$ then $F$ is irreducible and, in factabsolutely irreducible because we are handling coefficients in C. $\square$

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3 IIensel construction in $C\{y\}[x]$

The Hensel $consm_{1}ction$ is the p-adic univariate polynomial exnnsion technique devised by Hensel in $\sim 19(n$. and it is$gen\propto ahzedby$ Wang and Rothschild [WR75] to the expansion ofmultivariate polynomials. In this saeaon, we fornulatethe generalized version in $C\{y\}[x]$ . for simplicity.

We assume that $F(x, y)$ is a monic polynomial in $C\{y\}[x]$ and $F(x, 0)$ is square-free. Let $deg.(F)=n$ and the rootsof $F(x, 0)$ be $u_{1},$ $\ldots,$ $u_{n}$ ;

$F(x, 0)=(x-u_{1})\cdots(x-u_{n})$ , $u:\in C$ , $i=1,$ $\ldots,$$n$ . (3.1)

By assumption, $u_{i}\neq u_{j}$ for $i\neq!$ . We put

$F_{j}^{(0)}=x-u_{i}$ , $i=1,$ $\ldots$ , $n$ . (3.2)

Lemma 3.1 For integer 1, $0\leq l\leq n-1$ , there exists a set ofruunbers $\{w_{1}^{(l)}, \ldots, w_{n}^{(l)}\}sa\dot{a}\phi ing$

$w_{1}^{(l)}[F_{1}^{(0)}\cdots F_{n}^{(0)}/F_{1}^{(0)}]+\cdots+w_{n}^{(l)}[F_{1}^{(0)}\cdots F_{n}^{(0)}/F_{n}^{(0)}]=x^{t}$ . (3.3)

Proof Each term of the l.h.$s$ . of thhis equaUon is a polynomial of degree $n-1$ . Hence, we can determine $w_{1}^{(l)}$ ,$i=1,$ $\ldots,n$ , uniquely by evaluating the above equation at $n$ disunct points. Evaluating (3.5) at $x=u_{1},$ $\ldots,$ $x=u_{n}$ , wefind

$w^{(l)}:=u^{l}:/ \prod_{j-1,fl}^{n}(u;-u_{j})$ , $i=1,$ $\ldots,n$ . (3.4)

$\square$

If $H(x)=h_{n-1}x^{n-1}+\cdots+h_{0}x^{0}$, then (3.4) gives

$\sum_{\iota\triangleleft}^{n-1}w^{(l)}:h_{l}=H(u_{i})/\prod_{j-1,\prime j}^{n}(u;-u_{j})$ . (3.5)

Lemma 3.2 [generalized Hensel’s lemma]. Let $F(x,y)$ be defined as above. For any positive integer $k$ , there exists aset ofpolynomials $\{F_{1}^{(k)}, \ldots, F_{n}^{(k)}\}saa\phi ing$

$F(x, y)$ $\equiv$ $F_{1}^{(k)}(x, y)\cdots F_{n}^{\{k)}(x, y)(mod y^{k+1})$ , (3.6)

$F_{i}^{(k)}(x, y)$ $=$ $x-u_{t^{+C}:,1y+\cdots+C;,ky^{k}}$ , $i=1,$ $\ldots n$) ’ (3.7)

where $c_{i,j},$ $j=1,$ $\ldots,$$k$ , are numbers in C.

Proof By mathematical induction: Eqs.(3.6) and (3.7) are vdid for $k=0$. Assuming the validity of Eqs.(3.6) and (3.7),we consider the case of $k+1$ . Put

$F_{i}^{(k+1)}=F_{i}^{(k)}+\Delta F_{:}^{(k+1)}$ , $\Delta F_{i}^{(k+1)}\equiv 0(mod y^{k+1})$ , $i=1,$ $\ldots,$$n$ .

We determine $\Delta F_{i}^{(k+1)}.i=1,$$\ldots$ ) $n$ , so that they satisfy

$F=(F_{1}^{(k)}+\Delta F_{1}^{(k+1)})\cdots(F_{n}^{(k)}+\Delta F_{n}^{(k+1)})(mod y^{k+2})$

We can rewrite tluis equation as$\Delta F^{(k+1)}$

$\equiv$ $F-F_{1}^{(k)}\cdots F_{n}^{(k)}(mod y^{k+2})$. (3.8)

$=$ $\Delta F_{1}^{(k+1)}[F_{1}^{(0)}\cdots F_{n}^{(0)}/F_{1}^{(0)}]+\cdots+\Delta F_{n}^{(k+1)}[F_{1}^{(0)}\cdots F_{n}^{(0)}/F_{n}^{(0)}]$ .

By the induction assumption, the l.h.$s$ . of thhis equation can be expressed as$\Delta F^{(k+1)}\equiv y^{k+1}(f_{n-1^{X^{n-1}}}+\cdots+\tilde{f}_{0})(mod y^{k*2})$ . (3.9)

Hence, by Lemma 3.1, we can determine $\Delta F_{:}^{(k+1)}$ as.

$\Delta F_{i}^{(k+1)}=y^{k}:^{1}(\sum_{l\triangleleft}^{n-1}w_{1}^{(l)}f_{t})$ , $i=1,$ $\ldots,$$n$ . (3.10)

The $F_{i}^{(k+1)}$ constructed is of the form of (3.7). $\square$

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Lemma 3.3 (proofomitted). Let $F(x, y)$ $be$ &fned as above. Let$F(x, y)=G(x, y)\cdot H(x,y)$, (3.11)

where $G$ and $H$ are monic and elements of $C\{y\}[x]$ in general. Then, the Hensel construction of $F$ by the abovementioned method is equivalent to the Hensel constructions of $G$ and H. $0$

4 Algorithm of approximate factorization

We first consider power series by Hensel $cons\not\subset uc\dot{\mathfrak{a}}on$ . $p_{(r}$ simplicity, we confine ourselves to be in $C\{y\}[x]$ . Let $F’$ .$G^{(k)}$ and $H^{(k)}$ be elements of $C\{y\}[x]$ . If

$F’(x, y)\equiv G^{(k)}(x, y)H^{(k)}(x, y)(mod y^{k+1}\rangle$ , (4.1)

for any positive integer $k$ , we represent the case of $k=\infty$ as$F’(x, y)=G(x, y)H(x, y)$, (4.2)

where $G=1_{\dot{1}}m_{karrow\infty}G^{(k)},$ $H= \lim_{karrow\infty}H^{(k)}$ . We have seen in 3 that any polynomial $F(x, y)$ which is monic w.r $tx$

and for which $F(x, 0)$ is square-free, is factorized in $C\{y\}[x]$ as$F(x, y)$ $=$ $F_{1}(x, y)\cdots F_{n}(x, y)$ ,$F_{i}(x, y)$ $=$ $x-u_{i}+c_{i,1}y+\cdots+c_{i,k}y^{k}+\cdots$ , $i=1,$ $\ldots,$

$n$ , (4.3)

where $u_{i},$ $c_{i,1},$ $\ldots,$ $c_{i,k},$ $\ldots$ are numbers in C. In this and following sections, we define

$e=\deg_{y}(F)$. (4.4)

We want to calculate every irreducible polynomial divisor $G$ of $F$ , where $G\in C[x, y]$ , in terms of $F_{1},$$\ldots,$

$F_{n}$ . Thisis assured by the following theorem.

Theorem 4.1 (proofomitted). Let $F$ and $F_{1}.i=1,$$\ldots,$

$n$ , be defined as above. Any polynomial divisor ofF is aproductofelements of $\{F_{1}, \ldots, F_{n}\}$ , and each element $F_{i}$ is contained in only one irreducible polynomial divisor ofF. $\square$

Theorem 4.2 Let $\epsilon$ be a small positive number, $0<\epsilon\ll 1$ . Let F. G. $H$ be normalized monic polynomials in $C[x,y]$

such that $F(x, 0)$ is square-free and$F(x, y)$ $=$ $G(x, y)H(x, y)+\Delta F$,

(4.5)$F(x, O)$ $=$ $G(x, O)H(x, O)$, $mmc(\Delta F)=O(\epsilon)$ .

Let $F_{1}\ldots.,$ $F_{n}$ be defined as in (4.3). Then, some elements of $\{F_{1}, \ldots, F_{n}\}$ , let them be $F_{1}\ldots.,$ $F_{m}$ with $m<n$ . $sati\phi$

$G(x, y)=F_{1}(x, y)\cdots F_{m}(x,y)+\Delta G(x, y)$ ,(4.6)

$\Delta G(x, O)=0$ , $mmc(\Delta G)=O(\epsilon)$ .

Proof Put $F=G\tilde{H},\tilde{G}=G-\Delta G$ and $\tilde{H}=H-\Delta H$ , where$G=F_{1}\cdots F_{m}$ , $\tilde{H}=F_{m+1}\cdots F_{n}$ .

Substituting these into (4.5), we have$\Delta G\cdot H+\Delta H\cdot G+\Delta F=\Delta G\cdot\Delta H$ . (4.7)

Formula (3.10) shows that the correction terms $\Delta F_{:*}^{(k+1)}i=1,$$\ldots,$

$n$ , are linearly proporional to the coefficients of$\Delta F^{(k+1)}$ . This means that $\Delta F_{i}^{(k+1)},$ $i=1,$ $\ldots,$

$n$ , go to zero uniformly as $\Delta F$ goes to zero, and so are $\Delta G$ and $\Delta H$ .Hence, (4.7) means that

$O(mmc(\Delta G))=O(mmc(\Delta H))=O(mmc(\Delta F))=\epsilon$ .$\square$

If $G=F_{1}\cdots F_{m}$ is apolynomial divisor of $F$ , then $G\equiv F_{1}^{(k)}\cdots F_{m}^{(k)}(mod y^{k+1})$ , where $F_{:}^{(k)},$ $i=1,$ $\ldots$ , $n$ . are thefinite power series consmlcted by Hensel’s method. In order to test whether $G$ is an (approximate) divisor of $F$ or not,

we have only to set $k>e(=\deg_{y}(F))$ and perform the ffial division of $F$ by $G$ . Since $\deg(F_{i})=1,$ $i=1,$ $\ldots,$$n$ , we can

find all the irreducible ($a_{P\Psi^{oximate)}}$ factors of $F$ by $\ddagger esnng$ all the combinations of $F_{1},$$\ldots$ , $F_{n}$ successively from low

to high degrees.

This observation leads us to the following primitive algorithm which is applicable to $F(x, y, \ldots , z)\in C[x,y, \ldots, z]$ .

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Algorithm: Approx-Factorization (primitive versicn)

Input: a normalized monic and approximately square-free polynomial $F(x, y, \ldots , z)$,$\epsilon=the$ accuIacy of approximation, $0<\epsilon\ll 1$ .

Output: $\{G_{1}, \ldots , G_{r}\}$ , where each $G$; is an irreducible approximate factor, of acccuracy less than $\epsilon$ , of $F$ .

L Choose constants $y_{0},$ $\ldots,$ $z0$ such that $F(x, y0, \ldots, z_{0})$ is approximately square-free, and calculate the rootsof $F(x, y_{0}, \ldots, z_{0})$ numerically:

$F(x, y_{0}, \ldots, z_{0})=(x-u_{1})\cdots(x-u_{n})$.

$\Pi$. Put $S=(_{-}y-y_{0}, \ldots, z-z_{0})$, an ideal generated by $y-y_{0},$ $\ldots,$ $z-z_{0}$ . Using the generahzed Henselalgorithm, consmct $F_{1},$

$\ldots,$$F_{n}$ such that

$F(x, y, \ldots, z)\equiv F_{1}(x, y, \ldots, z)\cdots F_{n}(x, y, \ldots, z)(mod S^{e+2})$,where $e= \max\{\deg_{y}(F), \ldots, \deg_{z}(F)\}$ . $i=1,$ $\ldots n$)

$m$. Put $\Gamma:=\phi$ (null $s$et) and let $\tilde{\Gamma}$ be an ordered set of $aU$ the disunct factors of $F_{1}\cdots F_{n}$ , of degrees less thanor equal to $n/2$, arranged in low-to-high degree order:

$\tilde{\Gamma}:=(F_{1,}F_{n}, F_{1}F_{2}, \ldots, F_{n-1}F_{n}, F_{1}F_{2}F_{3}, \ldots)$ ,deg(element of $\tilde{\Gamma}$) $\leq n/2$.

While $\tilde{\Gamma}\neq\phi$ do( $G$ $:=$ [$the$ first element of $\tilde{\Gamma}$];ifmmc(remainder(F, $G)$) $\leq O(\epsilon)$

then $\Gamma:=\Gamma\cup\{G\}$ and delete all the multiples of linear factors of $G$ from $\overline{\Gamma}$ ,else $F:=F-\{G\}$).

IV. Remrn $\Gamma$ . $\square$

5 Efficient selection of the relevant combinations

The factorization algorithm given in 4 is, although complete, quite inefficient unless $n=\deg(F)$ is small. Thetme-consuming step is step $m$: in the worst case, we must check $2^{n-1}$ different combinauons of $F_{1,\ldots)}F_{n}$ . In thissection, we presenta simple and efficient method for finding the relevant combinations of $F_{1},$

$\ldots,$$F_{n}$ . Here, only abrief

description of the method is presented, and the detailed analysis will be given elsewhere. Fmhermore, we discuss in thuispaper only the properties of exact polynomial factors and the discussion from the viewpoint of approximate factonizationwill be given in another paper.

We assume again that $F$ is in $C[x, y]$ and, define $F_{1},$$\ldots,$

$F_{n}$ as in (4.3) and $e$ as$e=\deg_{y}(F(x, y))$ . (5.1)

The key point of our method is the investigation of the coefficients of higher degree terms in $F_{1},$$\ldots$ , $F_{n}$ . The existenoe

of such terms is assured by the following lemmas.

Lemma 5.1 (proofomitted). Let $F(x, y)\in C[x, y]$ and $F=GH$ . where $G$ and $H$ are monic and elements of $C\{y\}[x]$ .$IfG$ is afinuepower series in $y$ , or $G\in C[x, y]$ , then $G$ and $H$ are polynomial divisors ofF. (If $G$ contains a $temcy^{k}$ ,

$k>e$ . then $G$ is an infinite power series in $y$). $\square$

Our method of finding the relevant combinations is to utilize several kinds of linear relations satisfied by thecoefficients of the power series factors of polynomials. The first relation is as follows.

Lemma 5.2 Let $F$ and $F_{1},$$\ldots,$

$F_{n}$ be defined as in (4.3). If$G=F_{1}\cdots F_{m}=x^{m}+g_{m-1}(y)x^{m-1}+\cdots+g_{0}(y)$

is an element of $C[x, y]$ . then$c_{1,k}+\cdots+c_{m,k}=0$ for any integer $k>e$ . (5.2)

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Proof Put $F_{i}(x,y)=x-\varphi_{i}Cy),$ $i=1,$ $\ldots,n$ , then $\varphi_{1}Cy$)$\ldots.,$$\varphi_{m}(y)$ are roots of $G(x, y)=0$. Hence, from the

$reh\alpha onship$ between roots and coefficients, we have

$\sum_{:\sim 1}^{m}\varphi_{i}Cy)=-g_{m-\iota C}y)$ . (5.3)

$Obs\propto ving$ the $y^{k}$ -term of $\phi js$ equation, we obtain (5.2). $\square$

Terminology If $F_{1}\ldots..F_{m}sa\alpha sq(5.2)$ . we say $\{F_{1}, \ldots, F_{m}\}$ satisfies zero-sm relation. $0$

Let $M_{1}^{(e)}$ be the following $ma\alpha ix$ consrucoed from Ule coefficients $ofF_{i},$ $i=1,$ $\ldots$ , $n$ ;

$M_{1}^{(\epsilon)}=(\begin{array}{llll}c_{1,\epsilon*1} c_{1,\epsilon*2} c_{1,e*3} \cdots c_{2,e+1} c_{2,e+2} c_{2,e+3} \cdots| | | c_{n,e+1} c_{n,\epsilon+2} c_{n,e*3} \cdots\end{array})$ (5.4)

Then, Lemma 5.2 leads to the following theorem.

Theorem 5.1 The rank ofmatrix $M_{1}^{(e)}$ is equal to or less than $n-r$ , where $r$ is the number of irreducible polynomialdivisors ofF. $\square$

CoroUary. If the rank of $M_{1}^{(\epsilon)}$ is $n-1$ then $F$ is irreducible over $C$ , i.e., absolutely irreducible. $\square$

If $F$ contains irreducible factors of degree $\geq 2$, then some rows of $M_{1}^{(\epsilon)}$ are nonnull by Lemma 5.1 and we can findnonzero elements of $M_{1}^{(e)}$ by constructing $F_{:^{k}},$ $i=1,$ $\ldots$ , $n$ , up to $k\geq 2e$ . Therefore, in order to find the zero-sumrelations, we need not calculate infinite power series but have only to calculate finite terms of $F_{i},$ $i=1,$ $\ldots,$

$n$ .In many cases, we can find the relevant combinations of $\{F_{1}, \ldots, F_{n}\}$ by finding the zero-sum relations for row

vectors of $M_{1}^{(e)}$ . We ffist define non-overlapping relations.

Deflnition 5.1 [non-overlapping relations]. Let $v_{1},$ $\ldots,v_{n}$ be vectors in $C^{n’}$ , where $n\leq n’$ , and $R;a_{1}v_{1}+\cdots+a_{n}v_{n}=0$

and $R’$ : $a_{1}’v_{1}+\cdots+a_{n}’v_{n}=0$ be linear relations on $v_{1},$ $\ldots,$ $v_{n}$ , where $a_{i}\in C$ and $a’:\in C,$ $i=1,$ $\ldots$ , $n$ . $R$ and $R’\pi e$

caUed non-overlapping if $|a_{1}a_{1}’|+\cdots+|a_{n}a_{n}’|=0$ . $\square$

If there are $f$ linear relations which are linearly independent, then the relations form an r-dimensional vector space.Since the polynomial divisors of $F$ of the form $x-\varphi(y)$ can be found easily, we assume without loss of generality

that every row of $M_{1}^{(e)}$ is nonnull. Let $M_{1}^{(e)}=$ $(v_{1}, \ldots , v_{n})^{T}$ , then our problem is to find non-overlapping linear relationsof the form $a_{1}v_{1}+\cdots+a_{n}v_{n}=0$, where each $a_{i}$ is either 1 or $0$. The non-overlapping relauons can be found by thefollowing algorithm which finds $aU$ the linear relations first by neglecung the $condi\alpha on$ on $a_{i}$ and then decomposes therelations to non-overlapping ones if possible. Note that there may not exist such non-overlapping relations.

Algorithm: Find-Relations

Input: Vectors $v_{1},$ $\ldots,$ $v_{n}$ in $C^{n’},$ $n\leq n’$ ;Let $R$ be the vector space of linear relations on $v_{1},$ $\ldots,$ $v_{n}$ , and let $f=\dim(R)$ .

Output: If there exist $f$ non-overlapping relations $\{\tilde{R}_{1}, \ldots,\tilde{R}_{r}\}$ then $\{\tilde{R}_{1}, \ldots,\tilde{R}_{r}\}$ else NOT.

I. Find all the linear relations on $v_{1},$ $\ldots,$ $v_{n}$ by the Gaussian $elimina\dot{u}on$ .Let the relations obtained be

$R_{*}\cdot$ : $a_{i1}v_{1}+\cdots+a_{in}v_{n}=0$, $i=1,$ $\ldots,$$r$ .

II. Forn the matrix $A$ as follows:

$A=(\begin{array}{lll}a_{11} \cdots a_{1n}| \ddots |a_{r1} \cdots a_{rn}\end{array})$ $arrow R_{r}^{1}arrow..\cdot R$ (5.5)

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$m$. Eliminate the columns of $A$ , from the 1st to $(r-1)st$, by the Gauss method (Gaussian elimination), theneliminate the columns backward, i.e., from the rth to 2nd.Let the $resul\dot{a}ng$ matrix be $A$ which is of the form

$\lambda=(\begin{array}{llllll}\tilde{a}_{11} 0 \tilde{a}_{1,r+1} \cdots \tilde{a}_{1n} \ddots | \ddots |0 \tilde{a}_{rr} \tilde{a}_{r,r\star 1} \cdots \tilde{a}_{rn}\end{array})$ $arrow R_{r}^{1}arrow..\cdot R’$

,(5.6)

rv. If $\tilde{A}$ consists of mutuaUy non-overlapping row vectors then retum the row $vect\alpha s$, else retum NOT. $\square$

Theorem 5.2 Let $v_{1}\ldots..v_{n}$ and $f$ be defined as in algorithm Find-Relations. Then, the algorithm Find-Relationsis correct, i.e., it fn& all the non-overlapping linear relations on $v_{1},$ $\ldots.v_{n}$ provi&d that the number of suchnon-overlapping relations is $f$ .Proof After the Gaussian elimination in Step I, $aI1$ nonnull $vect\propto s$ calculated are linearly independent of each other,and $r=d\dot{m}(R)$ where $R$ is the vector space of linear relations on $v_{1},$ $\ldots$ , $v_{n}$ . Next, we consider the matrix $\overline{A}$ in (5.6).Note that $\tilde{a}_{11}\neq 0,$ $\ldots,\tilde{a}_{rr}\neq 0$ , because pivoting is made in Step I if neces$s$ary. We denote the $i$ th row of $\tilde{A}$ by $R_{1}’\cdot$ ,$i=1,$ $\ldots,r$ .

Suppose that there exist $f$ non-overlapping linear relations $\tilde{R}_{1},$ $\ldots,\overline{R}_{r}$ on $v_{1},$ $\ldots,$ $v_{n}$ . Then, we must have$\{R_{1}’, \ldots, R_{r}’\}=\{\tilde{R}_{1}, \ldots, R_{r}\}$ , where the propruonaUty constants are omitted. In order to see this, suppose, forexample, that $\tilde{R}_{1}$ : $\tilde{a}_{11}v_{1}+\cdots+\tilde{a}_{1\mathfrak{n}}v_{n}=0$ and $\tilde{R}_{1}\neq R_{1}’$ . Then, $\tilde{R}_{1}$ is expressed as $\tilde{R}_{1}=R_{1}’+b_{2}R_{2}’+\cdots$ , where at leastone of $b_{i},$ $i=2,$ $\ldots,$

$r$ , is not zero. Suppose $b_{2}\neq 0$, then $R_{1}$ is of the forn $R_{1}$ : $\tilde{a}_{11}v_{1}+b_{2}\tilde{a}_{22}v_{2}+\cdots=0$. Therefore,by assumption, $\tilde{R}_{2},$

$\ldots,$$R_{n}$ must not contain $R_{1}’$ and $R_{2}’$ so they must be expressed in terms of $R_{3}’,$

$\ldots,$$R_{r}’$ , which is

impossible because $\tilde{R}_{1},$ $\ldots,\tilde{R}_{n}$ are linearly independent of each other. $\square$

We show an example of factorization using the zero-sum relations for $M_{1}^{(e)}$ .Example 5.1 $F(x, y)=x^{4}+(y-2)x^{3}-(y+1)x^{2}+(y^{2}+2)x-y$ .By Hensel consffuction, we obtain

$F_{1}$ $=$ $(x-0)-y/2-y^{2}/8-y^{3}/16-5y^{4}/128+\cdots$ ,$F_{2}$ $=$ $(x-1)-y/2-y^{2}/8+0+y^{4}/128+\cdots$ ,$F_{3}$ $=$ $(x-2)+y/2+y^{2}/8+y^{3}/16+5y^{4}/128+\cdots$ ,$F_{4}$ $=$ $(x+1)-y/2+y^{2}/8+0-y^{4}/128+\cdots$ .

Coefficients of terms $y^{k},$ $k\geq 2$, give the mamx$-1/8$ $-1/16$ $-5/128$

$M_{1}^{(2)}$ $=$

$-1-11/1/_{/^{8_{8}^{8}}}^{/_{8}}000$ $-1_{0}1/_{0_{16}}^{0_{16}}1/_{0}^{/16}$ $-1-56/_{0^{28}}^{/_{0^{128}}^{1_{128}}}5/1/_{/_{128}^{128}}$

$\ldots\{\begin{array}{l}-v_{1}-v_{2}-v_{3}-v_{4}-v_{1}-v_{2}-v_{1}-v_{3}+v_{1}-v_{4}+v_{2}\end{array}$

We see that $\{F_{1}, F_{3}\}$ and $\{F_{2}, F_{4}\}$ satisfy the non-overlapping zero-sum relations, thus $F_{1}F_{3}$ and $F_{2}F_{4}$ are goodcandidates for polynomial divisors of $F$ . In fact,

$G_{1}\equiv F_{1}F_{3}=x^{2}-2x-y$, $G_{1}|F$,$G_{2}\equiv F_{2}F_{4}=x^{2}-yx-1$ , $G_{2}|F$.

$\square$

Finally, let us consider the case in which, although we may determine some zero-sum relations, the above-mentionedcolumn elimination $q$)$era\dot{n}on$ on $M_{1}^{(e)}$ does not give us other zero-sum relations uniquely or some zero-sum relationsare overlapping. In this case, we consider the zero-sum relations on the powers of roots. Let $\varphi_{i}(y)$ be the $i$ th root of$F(x, y)=0$:

$F_{i}(x, y)=x-\varphi_{i}(y)$ , $\varphi_{i}(y)=u_{i}-c_{i,1}y-c_{s,2}y^{2}-\cdots$ , $i=1,$ $\ldots,$$n$ . (5.7)

668

We calculate ($d_{:}l=2,$$\ldots,$

$n$ , and represent them as

$\varphi^{l}=c:,y^{k}+\cdots$ , $l=2,$ $\ldots,$$n$ . (5.8)

Without loss of genemlity, we assume that $G=F_{1}\cdots F_{m}$ is a polynomial divisor of $F$ . Then, $\varphi^{l_{1}}+\varphi_{2}^{l}+\cdots+\varphi_{m}^{l}$

can be represented by elementary symmetric polynomials in $\varphi\iota,$ $..,$ $\varphi_{m}$ and the degrees, w.r.$t$. $y$ , of such symmetricpolynomials are not greater than le. Henoe, we have

$\sum_{:-1}^{m}\lfloor\varphi^{l}:\rfloor_{le+1}=0$ , $l=2,$$\ldots,$

$n$ . (5.9)

Thus, we again have zero-sum relations. Since the above relations must hold for each value of $l$ independently, the aboverelations can be unified to the zero-sum relaUons on the combinations

$t_{1}\varphi;+t_{2}\varphi^{2}:+\cdots+t_{n}\varphi^{n}:$’ $i=1,$ $\cdots,$ $n$ , (5.10)

where $t_{1},$ $t_{2},$ $\ldots,t_{\mathfrak{n}}$ are arbimy parameters.

Let $\overline{M}^{(ne)}(t_{1}, \ldots,t_{n})$ be the following matrix:

$\overline{M}^{(ne)}(t_{1}, \ldots, t_{n})=(\begin{array}{lll}\sum_{l-1}^{n}t_{l}c_{1}^{(l)_{nc+1}} \sum_{l-1}^{n}t_{1}c_{1}^{(l)_{ne+2}} \cdots| \vdots \sum_{l=1}^{n}t_{l}c_{n}^{(l)_{ne+1}} \sum_{l\Rightarrow 1}^{n}t_{l}c_{n}^{\langle l)_{ne+2}} \cdots\end{array})$ . (5.11)

Theorem 5.3 If $F$ has an irreducible divisor in $C[x, y]$ , let it be $G=F_{i_{1}}\cdots F_{i_{n}}$ . then $\{F_{1_{\mathfrak{l}}}, \ldots , F_{i_{n}}\}$ satisfies the zero-sum relationfor $\overline{M}^{(ne)}(t_{1}, \ldots, t_{n})$. Conversely, $tf\{F_{1_{1)}}, \ldots F_{i_{m}}\}$ satisfies the zero-sum relation for $\overline{M}^{(ne)}(t_{1}, \ldots, t_{n})$ .then $G=F_{i_{1}}\cdots F_{1_{n}}$ is a polynomial divisor of $F$ .Proof The first hdf of the theorem is obvious from the above discussion, so we prove only the second half.By assumption, we see that for each $l,$ $1\leq l\leq n,$ $\varphi_{i_{1}}^{l}+\varphi_{i}^{l_{2}}+\cdots+\varphi_{i_{n}}^{l}$ is a polynomial in $y$ . By decomposing$\varphi_{1\downarrow}^{l}+\varphi_{i}^{t_{2}}+\cdots+\varphi^{\iota_{m}}:’ l=1,$

$\ldots,$$n$ , into elementary symmetric polynomials in $\varphi;_{1},$ $\ldots,$ $\varphi_{i_{n}}$ , we see that the symmetric

expressions are again polynomials in $y$ . Henoe, $aU$ the coefficients of $G=(x-\varphi_{i_{1}}(y))\cdots(x-\varphi_{i_{n}}(y))$ are polynomial$s$

in $y$ and $G$ must be a polynomial divisor of $F$ by Lemma 5.1. $\square$

Example 5.2

$F(x, y)=x^{6}-$ $(12-12y+3y^{2})x^{4}+(36-102y+66y^{2}-30y^{3}+9y^{4})x^{2}$

$(81-198y+247y^{2}-154y^{3}+49y^{4})$.

This example is constructed as $F=G_{1}G_{2}$, where

$G_{1}=(x-2+y)^{3}-(1+y+y^{2}+y^{3})$ , $G_{2}=(x+2-y)^{3}+(1+y+y^{2}+y^{3})$.

Let $\omega_{j}.j=1,$$\ldots,$

$6$ , be the six different roots of $x^{6}-1=0$:

$\omega_{j}=\cos(0-1)\pi/3)+i\sin(0-1)\pi/3)$ , $j=1,$ $\ldots$ , 6.

Thus, we have $\omega_{1}+\omega_{4}=0,$ $\omega_{2}+\omega_{5}=0,$ {$v_{3}+\omega_{6}=0$. Let $\overline{\varphi}$ be

$\overline{\varphi}$ $=$ $(1+y+y^{2}+y^{3})^{1/3}$

$=$ $1+y/3+2y^{2}/3^{2}+14y^{3}/3^{4}-46y^{4}/3^{5}+10y^{5}f3^{6}+\cdots$ .

Let the roots of $G_{1}(x, y)$ and $G_{2}(x, y)$ in $C\{y\}[x]$ be $\{\varphi_{1}, \varphi_{3}, \varphi s\}$ and $\{\varphi_{2}, \varphi_{4)}\varphi_{6}\},$ $res\mu ctively$ :

$\varphi j=2-y+v_{j}\overline{\varphi}$ , $j=1,3,5$ ,$\varphi_{j}=-(2-y)+\omega_{j}\overline{\varphi}$, $j=2,4,6$.

Thus, we have the relations$\varphi_{1}+\varphi_{4}=0$, $\varphi_{2}+\varphi_{5}=0$, $\varphi_{3}+\varphi_{6}=0$.

679

The $F_{:},$ $i=1,$ $\ldots$ , 6, constructed from $F$ by Hensel’s method are$F_{:}(x, y)=x-\varphi_{i}(y)$ , $i=1,$ $\ldots,$

$6$ .

Therefore, the rank of matrix $M_{1}^{(2)}$ is 1 and the row vectors of $M_{1}^{(2)}$ satisfy three zero-sum relations $v_{1}+v_{4}=0$ ,$v_{2}+v_{5}=0,$ $v_{3}+v_{6}=0$, as well as zero-sum relations $v_{1}+v_{3}+v_{5}=0,$ $v_{2}+v_{4}+v_{6}=0$ . However, only the $la\emptyset\epsilon r$ tworelations correspond to irreducible polynomial factors of $F$ and the former three relations do not. Note that the abovezero-sum $reIanons$ are overlapping. (Applying the column elimination procedure mentioned above to $M_{1}^{\{e)}$ , we obtain$w_{2}v_{1}-v_{2}=0$, w3$v_{1}-v_{3}=0,$

$\ldots,$$\omega_{6}v_{1}-v_{6}=0$ . These relations cannot be split into mutually non-overlapping linex

relations by algorithm Find-Relations.)

On the other hand, we have

$\varphi_{1}^{2}=\varphi_{4}^{2}=(2-y)^{2}+\omega_{1}(2-y)\overline{\varphi}+\omega_{1}\overline{\varphi}^{2}$,$\varphi_{2}^{2}=\varphi_{5}^{2}=(2-y)^{2}-\omega_{2}(2-y)\overline{\varphi}+\omega_{3}\overline{\varphi}^{2}$ ,

$\varphi_{3}^{2}=\varphi_{6}^{2}=(2-y)^{2}+\omega_{3}(2-y)\overline{\varphi}+\omega_{5}\overline{\varphi}^{2}$.

Hence, eliminating the columns of maffix $\overline{M}^{(3)}(t_{1}, t_{2},0, \ldots, 0)$, we obtain only zero-sum relations

$v_{1}+V_{3+v_{5}=0}$, $v_{2}+v_{4}+v_{6}=0$.

From these, we see that $G_{1}=F_{1}F_{3}F_{5}$ and $F_{2}F_{4}F_{6}$ are good candidates for polynomial divisors of $F$ , and in fact $G_{1}|F$

and $G_{2}|F$ . $\square$

6 Concluding remarks

In this paper, we have $in\alpha\alpha lu\infty d$ the concept of approximate factorization, presented a prmitive version of the algorithmof approximate factorization, and proposed a method which will make the algorithm practical. However, we have notfully analyzed the algorithm yet; a mathematical as well as computational analysis will be given elsewhere. In particular,we will show that finding the zero-sum $rela\dot{\mathfrak{a}}ons$ on the matrix $\overline{M}^{(ne)}(t_{1)}\ldots , t_{n})$ will determine the relevant combinationsof $\{F_{1}, \ldots, F_{n}\}$ uniquely to give all the irreducible polynomial divisors of $F$ [SSH91].

The basic principle of our factorization algorithm is quite simple, and we think the idea will be applicable to exactfactorization over integers, etc. We $a\infty$ now pursuing such possibilities, $t\infty$ .

6810

AcknowledgmentsThe authors thank Drs. T.Saito of Sophia Univ. and T.Hirano of Kanagawa Tech. Univ. for valuable discussions and Dr.K.Yokoyama of Fujitsu Ltd. for useful comments and references.

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